Does 6(x-9)=6x-54+x have infinite solutions?

Answer 1

To determine if the equation \( 6(x-9) = 6x - 54 + x \) has infinite solutions, we can simplify both sides and then see if they are equivalent.

First, simplify the left side:

\[ 6(x - 9) = 6x - 54 \]

Now, simplify the right side:

\[ 6x - 54 + x = 6x + x - 54 = 7x - 54 \]

Now we can rewrite the equation using these simplifications:

\[ 6x - 54 = 7x - 54 \]

Next, let's isolate \( x \) by moving all terms involving \( x \) to one side and constant terms to the other side:

Subtract \( 6x \) from both sides:

\[ -54 = 7x - 6x - 54 \]

This simplifies to:

\[ -54 = x - 54 \]

Now add 54 to both sides:

\[ 0 = x \]

This shows that \( x = 0 \) is the only solution. Since we have one specific value for \( x \), the equation does not have infinite solutions.

Therefore, the equation does not have infinite solutions; it has a unique solution \( x = 0 \).

Answer 2

Does 5(x-8)+10=5(x+2) have infinite solutions?

Answer 3

To determine if the equation \( 5(x - 8) + 10 = 5(x + 2) \) has infinite solutions, we can simplify both sides and see if they are equivalent.

Step 1: Simplify both sides

Left Side: \[ 5(x - 8) + 10 = 5x - 40 + 10 = 5x - 30 \]

Right Side: \[ 5(x + 2) = 5x + 10 \]

Step 2: Set the simplified expressions equal to each other

Now we need to set the left side equal to the right side: \[ 5x - 30 = 5x + 10 \]

Step 3: Isolate the variables

Next, we can see if there are any values for \( x \) that satisfy this equation. Subtract \( 5x \) from both sides: \[ -30 = 10 \]

This statement \( -30 = 10 \) is false, which indicates that there are no values of \( x \) that satisfy the equation.

Conclusion

Since we reached a contradiction and found that there are no solutions, the equation \( 5(x - 8) + 10 = 5(x + 2) \) does not have infinite solutions; in fact, it has no solutions.

Answer 4

A salesman makes both a base salary and also a commission, which is a percentage of what he sells. Each month, if his sales total s dollars, he makes a total of 2,000 + 0.1s dollars. What does 2,000 represent?

A. His monthly base salary in dollars
B. The amount of his monthly sales in dollars
C. His total monthly pay in dollars
D. His monthly commission in dollars

Answer 5

In the given expression for the salesperson's total earnings, \( 2000 + 0.1s \), we can analyze the components:

• \(2000\): This is a fixed amount that the salesperson receives each month, regardless of their sales.
• \(0.1s\): This represents the commission, which is 10% of his sales total, \(s\).

Given this breakdown, the \(2000\) in the expression represents the monthly base salary of the salesman.

Thus, the correct answer is:

A. His monthly base salary in dollars.

Answer 6

The value of a baseball card in dollars has been found to be 0.15y + 0.35, where y is the number of years since it was released. By how much is the baseball card's value increasing per year?
A.
$0.15

B.
$0.35

C.
15%

D.
35%

Answer 7

The expression for the value of the baseball card is given as:

\[ V = 0.15y + 0.35 \]

where \( V \) is the value of the baseball card in dollars, and \( y \) is the number of years since it was released.

To find how much the baseball card's value is increasing per year, we can look at the coefficient of \( y \) in the equation. The coefficient \( 0.15 \) represents the increase in value for each additional year.

Thus, the baseball card's value is increasing by:

\[ \text{$0.15 per year} \]

Therefore, the correct answer is:

A. $0.15.